On transformation of infinite products

ON TRANSFORMATION OF INFINITE PRODUCTS

SOON-Mo JUNG

1. Introduction

In the classical analysis there are various theorems which permit us to interchange limits and infinite sums, limits and integrals, integrals and infinite sums, etc. The infinite products as well as the infinite series play an important role in different branches of mathematics. It is therefore natural tp study the conditions under which the infinite products and the infinite (or finite) sums are interchangeable. But there is unfortunately no theorem which permits us to do so, although many authors have investigated the infinite product identities (see [1], [2], [3], [6] and [13]) and the convergent properties of power product expansions of analytic functions (see [4], [5], [7], [8], [9], [10], [11], [12]

and [14]).

For any rn, n E N we define

J: ^{={(ji)i=l, ,m : ji =1, ... ,n}, In =}{(ji)i=1,2, : ji =1, ... ,n},

J: ^{={(ji)i=l, ,m : ji =1,2, ...},Joo =} {(ji)i=1,2, : ji =1,2, ... }.

Empirically, according to the distributive law, we know that if aij are real numbers then

(all +^a12+^a13)(a21 +^a22+^a23)=^alla21+^alla22 +^{an a23} +^{a12a 21} +^{a12a 22}+^a12a23+^a13a21 +^a13a22+^a13a23,

I.e.,

2 3 2

IT^{Laij = L} IT^aij;'

i=lj=I (j;)EJii=l

Received October 12, 1994.

1991 AMS Subject Classification: 40B05.

Key words and phrases: Infinite product, transformation.

We can spontaneously generalize the above equation as follows

n m n

(1.1) ITI:^{aij =} I: IT^aiji

i=1 }=1 (j;)EJ;::. i=1

when ail are real numbers. Furthermore, it is not difficult to verify

n o o n

ITI:^{aij =} I: IT^aiji

i=1 }=1 (ji)EJ;:' i=1

if the series E~1^aij (i =1, ...,n) converge absolutely.

We may now raise some questions whether the following equations

(1;2)

00 n 00

-HL a i j = - E- n^{aili ,}

i=1 j=1 (ji)EJn i=1

(1.3)

00 00

I T L : a i j =

i=1 j=1

00

I: IT^aij;,

(ji)EJ"", i=1

which are generalized forms of the last equation, are also true.

DEFINITION. An infinite product IT:1ai is called convergent, if there is a number n E N such that limN->oorr~n ^ai exists and is different from O. Otherwise the infinite product is called divergent.

Suppose that for alli EN and for j =^{1, ... ,n (n ~ 2)}

-1 + ^-1'-2

aij = n- n z •

Then we have for all (ji) ^E Jn

n00 ^{aiji =} 0 (divergent to 0).

i=1

Hence

I: n00 ^{aiji = O.}

(ji)EJn i=1

However, we obtain

00 n 00

IT^{L aij =} IT(1+^i-2) ~ 1,

i=1j=1 i=1

I.e., the infinite product n:1 (~j=1^aij) converges unconditionally and

00 n 00

ITLaij =I L IT^aiji'

i=1j=1 (ji)EJ"i=1

Equation' (1.2) is not trivial in .such a sense that not every double sequence (aij);=1,2, ...;j=1,...,n satisfies (1.2) (for convenience, we call (aij)i=l,2, ...;j=1,...^,n a double sequence).

Now let (aij)i,j=1,2, ... be a double sequence satisfying .. - 6 -2 '-2 (1+3-1 '-2)

a_{lJ -} ^7r J Z .

IT00 ^{aiji =} 0 (divergent to 0).

;=1

Therefore

00

L IT^{aiji = O.}

(ji )EJooi=l On the other hand,

00 00 00

IT^{L aij =} IT^{(1+3-1i-2) ~ 1.}

;=1 j=l i=l

This example implies that the equation (1.3) is not trivial.

In this paper, we shall find some sufficient conditions under which (1.2) holds true and treat the equation (1.3).

2. Transformation of infinite products with finite sums as their terltlS

Let p > 1 be a fixed real number. Suppose (Si) to be a monotone increasing sequence such that there exists a positive number ^0: with

Si ~ iI+ofor all i E N. Let MllM₂be positive numbers with Ml ~ M₂ and let (ti) be a bounded sequence satisfying ti > 1 for all i E Nand

n:l^{ti = M}l-^{For j =} 1, ... , n let (aij) be a real sequence with the following properties:

00

(2.1) 0 < ail::; ti for all i EN and IT^{ail = M2 ,}

i=l

(2.2) IaijI ::;p-Si (j = 2, ... , n) for sufficiently large i EN,

··(2~31

00 n

·L:"TL"laiij~lr<oo .

i=l j=l

EXAMPLE 1. The double sequence (aij) with for j = 1, forj =^{2, ...,n}

satisfies the conditions of (2.1), (2.2) and (2.3).

THEOREM 1. It holds that

00 n 00

IT^{Laij = L} IT^aij;·

i=l j=l (ji)EJ"i=l

Proof For any N E N let N n

AN = IT^Laij,

i=l j=l

N n

A~ ⁼ IlL ^laijl,

i=l j=l

N

BN = L IT^aiji'

(ji)EJ~i=l N

B~⁼ L IT I^aijiI.

(ji)EJ~i=l

Then, by (1.1), AN =B N and AN =^BN^{for all N E N. Hence}

(2.4) By (2.3)

. (2.5) lim AN <⁰⁰ and lim AN < ^{00 •}

N--= N--=

Combining (2.4) and (2.5), it is enough to prove that

(2.6)

= lim BN = L IT^aiji'

N--= (ji)EJn i=l

where we assume that the summation over I_n is an ordered sum, e.g., In may be lexicographically ordered. For any sufficiently large N EN let

(2.7)

= = =

aN = L IT^{aiji +} L IT^{aiji +} L IT^aiji

ji=l fori=l ji=l fori=l 3i$N:j.>1 i=l alli>N all i$N 3i>N:ji>1

N = = =

= L IT^aij. IT ^{ail +} L IT^{aij. +} L IT^aij;

ji=l fori=l i=N+1 ji=l fori=l 3i$N:ji>1i=l

all i>N all i$N 3i>N:ji>1

=

=BN IT ^{ail +ak, +aJ..,}

i=N+1

be a reordering of E(j;)EJ_n n:l^aij" where the :first term includes

n:1^{ail =} M2 , and similarly, let (2.8)

=

aN = L IT^{IaijiI +} L IT ^{IaijiI +} L IT ^Iaij;I

ji=lfori=l ji=lfori=l 3i$N:ji>1 i=l

all i>N all i$N 3i>N:ji>1

=BN ITI^ailI +a~+ _a~

i=N+l

be a reordering of L(i;)EJnn:1 ^{1aij; I,} where the first term includes

n:1 ^{lai11· Withq = sup{i >N :ii > I} and r = #{i >N :ii > I},}

assume that the terms ofO'~are so ordered that the following estimate is possible:

I"

QC q-N

O'~ ~ L L L ^{M 1}II ^landn;I

q=N+1 r=1 N<nl <···<nr=q i=l 2Sintsnfor t=l, ...,r

by (2.1)

I"

QC q-N

~M1 L L L ^IIp-sn;

q=N+1 1"=1 N<nl <"'<nr=q i=l 2Sintsnfort=l,...,r

by (2.2)

QC

=M¹ L

··4J=N+1 q-N-1

1"=0L

I"

L ^P-Sq II^p-Sn;

..N<nl<·.···<nr-<q i=1.

2Sintsnfort=l,...^,r

QC

=M1 L ^{P-Sq (1+(n} _1)p-sN+l )q-N-1 q=N+1

QC

~M1 L ^{P-Sqexp((n -1)(q - N _1)p-SN+l)}

q=N+l (2.9)

(2.10)

-+0 as N -+ 00.

I" QC

M¹II ^{1andn; 1} L II

i=l i;=lfori=N+l

alliSN

With the notation {i _~ N :ii > I} = {nt, ... , nr } we suppose that the summation ofO''J is such an ordered sum that the following estimate is allowed:

N 0"2_{N -}<~L...J

1"=1 l<nl<"'<nr<N 2si-;'tsnfor t=I,...,r

By (2.2) there exists a positive number M such that I^aij I :::; ^{M for}

j = 2, ...,n and for all i E N. Since N is sufficiently large, we have by (2.1), (2.9), (2.10) and the ratio test for convergence of series

u~ ^:::;Ml ~ ((~)(n -ltMr2M;lu~)N

:::;2MlM_2- ^l ((1 +(n -l)M)N-1)u~

:::;2M;M;l(1 +nM)N_p-N¹+o./²

(2.11)

~00 _(ql+o._N¹+o./^{2 ) (} _N¹+o.)

L...J p exp nqp

q=N+l

~o as N ~ 00.

Combining (2.8), (2.10), (2.11), (2.4), and (2.5)

00

lim u~ ^{= lim} B~ IT ^{laill = lim B~ < 00,}

N-oo N-oo N-oo

i=N+l

since n : N + l I^ail I~ 1 as N _~ 00. Hence the series UN converges unconditionally ifN is sufficiently large. Therefore

(2.12)

:E IT00 ^{aiji = UN}

(j;)EJn i=l

for sufficiently large N. On the other hand, by (2.10) and (2.11) (2.13) lim u1^{= 0 and lim} u'Jv ⁼ o.

N-oo N-oo

Itfollows then from (2.7), (2.12), and (2.13) that

00 00

L IT^{aiji =} _N-oo^{lim BN} IT ^{ail =} _N-oo^{lim BN,}

(ji)EJn i=l i=N+l

as required.

3. Interchange of infinite products and limits

Let (aij) be a double sequence satisfying aij = 671'"-2j-2 for each i E N. Since _E~lj-2 - 6-¹71'"2, we have for any n E N and for every i EN

n 00

L aij = 1 - 611"-2 L j-2 and

j=l j=n+1

Hence

00 n

00 00

L671'"-2 L ^{j-2 = 00.}

i=l j=n+1

IILaij = O.

i=l j=l

Therefore

00 n

lim II ""'^{aij = O.}

n--+oo L..J i=1j=1

On the other hand,

00 00 00

II^{L aij =}II^{1= l.}

i=lj=l i=1

In this case the infinite product symbol and the limit symbol cannot be interchanged, i.e.

o o n ^{0 0 0 0}

n1!...~II^{L aij #}II^{L aij'}

i=lj=l i=1j=1

In this section we find some sufficient conditions under which the infi- nite product symbol and the limit symbol can be interchanged.

Let (sn) be a positive sequence with Sn -+ 00 as n ^{-+ 00} and let (aij) be a non-negative double sequence with the following properties:

(3.1)

00

Vi E N 3ri > 1 : 0~ ^{L aij} ~ ^ri,

j=1

(3.2) 3M>1 IIri^=M,

i=1 (3.3) 3a >0 3mE N Vi EN Vn ~ m

EXAMPLE 2. The double sequence (aij) with for j = 1, for j = 2,3, ...

satisfies (3.1), (3.2) and (3.3).

LEMMA 2 (CF. [14], THEOREM 2). Let (aij) be the double se- quence described above. Then

oon ^{0 0 0 0}

n~~IILaij=^IILaij.

i=lj=l i=lj=l

Proof Itfollows from (3.1), (3.2)and(3.3) that for sufficiently large Nand n ~ m

N 00

TIEaij

i=l j=l

+M_it t ... t1=1 i2=il+1 i N =iN _ 1+1 ^(f:j=n+1^{aid) (};=n+1^{f: ai2j ) ....• (} j=n+1^{f: aiNi)}

N n 00

STI E aij^+ME(8~ _1)-n(n+1)/2 .

i=lj=1 i=l

Byletting N ~ ⁰⁰it follows that

<Xl <Xl <Xl n

IT'E^aij ~ IT'E^{aij +}O(s~a)

i=l j=l i=l j=l

and then by letting n ~⁰⁰ again

<Xl 00 <Xl n

I I ' E a i j ~ n~~IILaij.

i=l j=l i=l j=l

The opposite inequality is obvious, since(aij) is a non-negative double sequence.

4...Transfol"mation()finfinit~produ~tswith infinite SUIIlS

as their terms

Let p > 1 be fixed. Suppose (Si) to be a monotone increasing sequence such that there exists a positive number a with Si ~ ^iI+a for all i E N. Let Mt, M2 be positive numbers with M1 ~ M2 and let (ti) be a bounded sequence satisfying ti > 1 for all i E N and

n:1^{ti = M 1 • Let (aij)} be a non-negative double sequence with the following properties:

00

(4.1) 0 < ^ail ~ ^ti for all i EN and IT^{ail = M 2 ,}

i=l

(4.2) 3{3 >0 Vi _~ 1 Vj >1 : 0 _~ aij ~p-P8d,

00 00

(4.3) L l'Eaij-11 ^{< 00.}

i=l j=l

EXAMPLE 3. The double sequence(aij) with for j =^1,

for j = 2,3, ,., satisfies (4.1), (4.2) and (4.3).

THEOREM 3.

00 00 00

IT^{L a i j =} 2~~ ^L IT^aiji·

i=l j=l (ji)EJn i=l

Proof For all sufficiently large n E N let

00 n

An =I T L a i j i=l j=l

00

and Bn = ^L IT^aiji·

(ji)EJn i=l

It follows from (4.2) and (4.3) that for all sufficiently large n E N

00 n

L I L a i j -11 < ^00.

i=l j=l

Therefore, it follows from Theorem 1that An = B n^{for all} sufficiently large n E N. So

(4.4) _n--+oolim An = lim Bn.

n~<::X)

On account of Lemma2, (4.4) and(4.3)

00 00 00

IT "'"_L...J^{aij =}n--+oo^{lim An =}n-+oo^lim "'"L...J IT^{aiji < 00.}

i=l j=l (ji)EJn i=l

Open problem. Under the conditions described above, does it hold true

00 00

J~~ ^L IT^{aiji = L} IT^{aiji ?}

(ji )EJni=l (ji)EJooi=l

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